Ramsey numbers of trees and unicyclic graphs versus fans

Abstract

The generalized Ramsey number $R(H, K)$ is the smallest positive integer $n$ such that for any graph $G$ with $n$ vertices either $G$ contains $H$ as a subgraph or its complement $\overline{G}$ contains $K$ as a subgraph. Let $T_n$ be a tree with $n$ vertices and $F_m$ be a fan with $2m + 1$ vertices consisting of $m$ triangles sharing a common vertex. We prove a conjecture of Zhang, Broersma and Chen for $m \ge 9$ that $R(T_n, F_m) = 2n - 1$ for all $n \ge m^2 - m + 1$. Zhang, Broersma and Chen showed that $R(S_n, F_m) \ge 2n$ for $n \le m^2 -m$ where $S_n$ is a star on $n$ vertices, implying that the lower bound we show is in some sense tight. We also extend this result to unicyclic graphs $UC_n$, which are connected graphs with $n$ vertices and a single cycle. We prove that $R(UC_n, F_m) = 2n - 1$ for all $n \ge m^2 - m + 1$ where $m \ge 18$. In proving this conjecture and extension, we present several methods for embedding trees in graphs, which may be of independent interest.